I was wondering if anyone can tell me the strength of the force between two massive objects according to GR. For the sake of simplicity, let's assume that both objects are perfect uniform spheres, and the mass of one object increases from the mass of an electron to the mass of a blackhole, while the other mass remains constant (1 kg). The distance between the two masses is 100,000 m.

I generally need to know:

1) Is the force between to masses in a microgravity field smaller or larger in GR than it is in Newtonian gravity. If it is smaller or larger, than by how much?

2) Is the force between to masses in a macrogravity field smaller or larger in GR than it is in Newtonian gravity. If it is smaller or larger, than by how much?

I assume that in standard gravitational fields, the gravitational force in GR and the classical model are equal since physicists are still using Newton's formula. If they are not completely equal, by how much are they different?

Tom

Lets not talk about force, but of potential energy instead. Of course one can be derived from the other, its just that most deal with energy.

The potential energy of a mass in a gravitational field, in the Newtonian picture is given by

Vn=-GmM/(r)
Vn is newtonian potential energy, G is the gravitational constant, M is the mass of say your 1kg mass, m is the other mass and r is the distance between the 2 bodies centres.

I intended to put something down for einsteins formulation, but it is not as simple as writing down an equation as per the newtonian case. I will check my sources in fear of putting don something incorrect.

Tom,

Again you are going for a hypothetical situation.

M1 is increasing from mass of one elctron to the minimum mass required for a black hole, Mbh.

1. The gravitational force between them increases with increase in M1. the distance between M1 and M2 (1kg constant) will not remain constant (100 km).

2. Long before M1 attains the Mbh, M2 would become part of M1.

You are left with only one mass, M1 that somehow increases constantly?? Please be more specific.

I'd just like to say that I am glad that we have moved onto more advanced topics:)

everneo,

I just want to know the force between the two objects if the mass of one of the objects increases while the distance between the objects remain constant.

For the sake of arguement, assume that there is a force keeping the distance constant.

Tom

I) Newtonian gravity is expressed as

Gravitational force = (G * M₁' * M₂) / r²
= G*M₁' * 1 / r² ---------> (1)

where G is the Gravitational constant, M₁' is the current mass of M₁; M₂ = 1kg and r is the distatnce between the centres of gravity.


II) GR does not treat the gravity as force but accelerated motion of M₂ in space-time, curved by the increasing mass of M₁.


M₂a = M<sub>2</Sub>dv/dt
= M₂[ (dvⁱ/dt)e_i + vⁱ(de_i/dt) ] ----> (2)

where

a = dv/dt = first derivative of velocity vector v and v = vⁱe_i ;

vⁱ is the components of velocity and
e_i basis (unit) vectors.

If you want to apply GR expression for inertial,cartesian coordinate system , vⁱ(de_i/dt) in (2) vanishes and

M₂a = M₂(dvⁱ/dt)e_i -----------> (3)

You can take any of the above 3 to get gravitational force (!) between the 2 bodies. (2) and (3) require a little familiarity with tensors.

Originally posted by ryans
I'd just like to say that I am glad that we have moved onto more advanced topics:)

I guess it's a mark of true intelligence pointing this out so eagerly :)

Tom:

The simple answer to your question is: there is no force between 2 massive objects in general relativity. Gravity is not a force in GR.

James R: Somebody here said gravity does work when I asked. If gravity is no force, how can it do work?

If it doesn't, could gravity, theoretically, be used to accelerate something to Light Speed?

everneo and James,

I was hoping that in the simplified example I have illustrated, that I might find the force, or acceleration, between two objects according to GR without having to use tensors.

Can someone show me the equation that I can use to find the difference in acceleration between the GR model of gravity and Newton's model of gravity in the example I provided. I need to to know exactly how the results from GR differ from Newtonian gravity.

Also, can someone explain, generally, how Einstein derived his gravitational formulas for GR. What did Einstein use to determine the amount of curvature of spacetime? Are his formulas based on observation?

Tom

Prosoothus,

I have no idea if it was so but I read one paper that claimed Einstein reverse engineered the solution. That was started with the orbit of Mercury and worked backwards to find what was needed to make the numbers fit the observation.?

If Einstein formulated GR, by reverse engineering (that requires the lot of knowledge ironically), to explain the unaccounted 40" in advance of mercury's perihelion by 2' every century then how come GR could also predict this ?

"Einstein's calculations in his newly developed general relativity indicated that the light from a star which just grazed the sun should be deflected by 1.75 seconds of arc. It was tested during the total eclipse of 1919 and during most of those which have ocurred since."
http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/grel.html#c2

If you could reverse engineer from a natural fact to formulate a theory that not only explains that fact but also other entirely different facts then you deserve Nobel Prize.

Before throwing dirt on Einstein please read the history of GR if not GR.

So by newton gravity is a force.

But in GR, Quoting you :- In the general relativistic picture of gravity, the "natural" state of motion for objects is free-fall. Things fall unless a force stops them doing that. Gravity itself is not a force, but the curvature of spacetime which determines which way things fall.

So if I am getting to grips with this, the newtonian model is a 3D geometry and the GR model is a 4D geometry.

To see gravity in 3d you would have to animate it, to see it in 4D you could view it as a 3d model so long as your brain understands it as a 4d model? A curve or worldline representing a path of a particle.
a 3d model of 3d geometry would show you static particles.?
a 3d model of 4d geometry would show you the path of a particle but not the particle itself.?

Actually that does not make sense because if gravity is an acceleration then in the 4d model the curve would only represent linear motion.

Anyway the original question should be answered better, maybe rewritten as :- if the bodies were to be allowed to collide by the action of gravity would both models show the colision occuring in the same ammount of time, or do they show the ammount of time to be different, which model shows the shortest ammount of time.

AndersHermansson:

<i>Somebody here said gravity does work when I asked. If gravity is no force, how can it do work?</i>

Are we talking Newtonian physics here, or GR? In Newtonian physics, gravity is a force and it does work. In GR it does no work because it is not a force.

<i>could gravity, theoretically, be used to accelerate something to Light Speed?</i>

No. The thing would need to accelerate for an infinitely long time.


MacM:

<i>I have no idea if it was so but I read one paper that claimed Einstein reverse engineered the solution. That was started with the orbit of Mercury and worked backwards to find what was needed to make the numbers fit the observation.?</i>

Now you're trying to downplay one of the best examples of deep thought in the 20th century. Since you know nothing of GR, you have no idea what it would take to reverse engineer it from the perihelion advance of Mercury. I would venture to suggest that such a thing is actually impossible. In any case, that is very far from what Einstein did. Even if you can't be bothered learning about the theory, the history is easy to check.